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Using Inequalities to Solve Equations

Suppose we need to solve equation `f(x)=g(x)` and suppose there exists such number A that is the maximum value of function `y=f(x)` and minimum value of function `y=g(x)`.

Then roots of the equation `f(x)=g(x)` are common roots of the equations `f(x)=A,g(x)=A` and only they.

Example. Solve `x^2+1/x^2=2-(x-1)^4`.

If we directly begin solve this equation, then we will end up with the equation of 6-th degree that is not very pleasant to solve. So, we need some trick here.

Note, that by Cauchy's inequality `(x^2+1/x^2)/2>=sqrt(x^2*1/x^2)` or `x^2+1/x^2>=2`.

From another side, since `(x-1)^4>=0` then `2-(x-1)^4<=2`.

This means that roots of initial equation are common roots of equations `x^2+1/x^2=2` and `2-(x-1)^4=2` (if there exist some roots).

From equation `x^2+1/x^2=2` we obtain that `x_1=1,x_2=-1`.

Frrom equation `2-(x-1)^4=2` we obtain that `x=1`.

Common root of these two equations is `x=1`. Therefore, `x=1` is the only root of the initial equation.